I usually don't get too many major injuries. in 4 years I had only one recruit that suffocated on the first year, and 2 swordsmen with minor spinal injuries. I currently have 3 spearmen and 8 swordsmen.
As for training methods, I usually start training my military before I have a metalsmithing operation prepared. Supply the recruits with leather armor, leggings and shields as soon as I can, add some turtle shell stuff for most of the other spots, and then by the time my metalsmithing operation is prepared they are usually decent wrestlers and have some armor and shield skill.
If I have silver I start making some silver weapons so they begin training their weapon skills, the silver should still not be able to damage much through the leathers, and if I got a moody bonecarver by then, the turtle shell junk should already be almost as good as iron. If I don't have silver I work on some metal armor, starting with chainmail and greaves to add to the leather, then make the weapons. I rarely see internal wounds, and those that I do see usually aren't fatal. Even if they have a spinal injury they're still soldiers, they just stopped training. And since they stopped training I can safely give those guys my big dangerous weapons.
As for the champions causing more injuries, Toady did mention that higher skill is less chance to accidentally make a lethal strike. When recruits make a lethal strike, even if more common, they don't have much damage behind it, because they are unskilled and have few if any attributes. But champions on the other hand when they eventually do make a lethal strike while sparring, which is inevitable, they have legendary skill behind it and probably a few points of strength too. Making their hits far more devastating.